: Sistem Informasi Penelitian Universitas Kristen Satya Wacana M02349

Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

INNOVATING MATHEMATICS
BY RESEARCH APPROACHES IN INDONESIA
FROM MIDDLE SCHOOLS TO UNDERGRADUATES
Hanna Arini Parhusip
Mathematics Departement –Science and Mathematics Faculty
Universitas Kristen Satya Wacana
www.uksw.edu

Abstract. Innovating mathematics by research approaches in Indonesia from middle schools
to undergraduates are explained here where geometry is typical example to be innovated.
Topics in plane geometry to be platonic solids and non platonic solids such as cylinder and
spheres are innovated to be new surfaces and media. Topics in calculus and complex
functions particularly parametric curves are used to be new curves for curves stitching,
motifs in several kinds of souvenirs and accessories. Additionally, local culture is introduced
through batik painting activity using algebraic surfaces drawn with Surfer. Leonardo Dome
and Leonardo Bridge with Rinus Reolofs like are introduced to attract students for

mathematical sports activities.
Keywords : plane geometry, parametric curve, complex function, Leonardo Dome, Leonardo
Bridge, batik, Surfer.

1. INTRODUCTION
When I come to the word innovation in mathematics, it turns out that an innovation is
a new method or branch in mathematics itself since ancient time. One has proposed the top
ten of innovation in mathematics enriching branches in mathematics such as Logarithms,
Matrix algebra, Complex numbers, Non-Euclidean geometry, Binary logic Decimal fractions,
Zero and Negative numbers Calculus and Arabic numerals. On the hand, an innovation in my
point of view is doing a new creativity from the classical part of a special issue in
mathematics and treating it to be new performance or new application with integrating
other approaches. Additionally, since technologies are growing in this 21st century, rapid
innovations in mathematics vary heterogeneously depending on the level of a society to
adjust the technologies such that trends on innovating mathematics may differ for each
different place or country.
Current trends on innovating mathematics in 21st century is merely its application in
science (Web1) such that curriculum must be redesign (Mahajan,2014) containing the
knowledge, skills and character. In the mathematical knowledge, one must know traditional
and modern knowledge requiring multidiscipline. In the skills ability, students are required

to know how to use leading to creativity, critical thinking, communication and how to
collaborate. Finally, a student must learn how to engage in the world with mindfulness,
curiosity, courage, resilience, ethics and leadership. Thus, an innovation on mathematics will
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Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

vary due to many aspects above. Several approaches will be recalled here to present for the
readers how to design creativity in some topics in curriculum and students are engaged with
their own innovations and creativities.
Innovating mathematics in this paper addresses on how presenting mathematics
differently compared to regular meeting in a classroom leading to some interactions among
students, educators and other necessary agents to collaborate with. As a result, new
materials may be necessarily prepared. The activity involves not only a single topic in a
curriculum but also integration of several topics that may appear due to an innovative
sense.
2. RESEARCH APPROACH FOR OLD AND MODERN GEOMETRICAL TOPICS

There exists no standard theoretical parts for innovating mathematics since the
innovations depend on which basic knowledge will be innovated, i.e. the regular formulation
and ideas in mathematical topics are explored into new performance, new beneficial and
new audience. To gain a maximal innovation, an integration of several topics is necessary.
The contents below are mostly few examples where middle or high schools and
undergraduate mathematical topics are studied and innovated by research approaches.
2.1 Research approach in geometry of platonic solid
In traditional approach, one must learn some basic knowledge in geometry, e.g.
definitions of several distances (between a point and line, point to a plane, a line to a plane,
and between 2 planes) where this knowledge is considerably neglected by most students. In
fact, students could not make the definitions on their owns though the situations are very
regular in daily life. The attitude for delivering mathematical definition without
communicating it needs leads to no critical thinking and ability to communicate. Instead,
students are posed several new creation and innovation with geometrical old patterns as
illustrated in Fig 1. Students learn about platonic solids and the properties leading to an
innovation of dress created by students which make students proud to promote.

Figure 1. Platonic solids in traditional view (left) and in the new performance (middle and right) created by
undergraduate mathematics education students in UKSW Indonesia , Sep 2016.

Additional approach can be done by using available software for drawing such as Geogebra
since students may show several mathematical properties for each solid.
Rinus Roelofs has more prominent innovations with platonic solids and more
innovated geometries as a sculpturist. Roelof’s i trigui g s ulptural o je ts are ade ith
all kinds of materials including in addition to paper wood metal and acrylic (Web 2). One
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Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

may find by search engines to overview many kinds of his innovations in mathematical
designs to approach a geometry lesson into an interesting topic. Fig.2 depicts several
elevated polyhedral of Rinus Roelofs like.

Figure 2. Elevated polyhedral created by Rinus Reolofs (source : private photo , taken from IC16, Berlin,
Germany, July 2016).

2.2 Research approach in Calculus and Complex Functions

Mathematical functions in calculus and complex functions normally are introduced
traditionally, i.e. dealing with computation and technical standard for drawing functions.
Though introducing functions can also be visualized with software (e.g. Geogebra, MAPLE),
the benefits or learning those topics are too formal. Instead, special topics in calculus and
complex functions can be combined into a creative activity and the example is shown here.
In high schools level, students may create the equations with excel and innovate the
visualization into objects. As stated before, the innovation may create a similarity with
nature. Let us consider the parametric equations read as (Parhusip,2016)
x  sin 2 (4 )  cos( 4 ) cos 





y  sin 2 (4 )  cos( 4 ) sin 

which are obtained from polar equation, i.e.

r  sin 2 (4 )  cos( 4 ). .

(1)

The obtained plane curve is similar to a flower called Icora Javanica and visualized in Fig.3
Mapping with complex function f(z)=exp(z), puzzle and lampshade are designed and
depicted in Fig. 3.
A parametric curve called hypocycloid curve has been modified in various types by
considering the resulting curves to be domains for complex functions
(Suryaningsih,et.al.,2013) (Parhusip,2014)(Purwoto, et. al.2014) (Parhusip,2015). Results on
souvenirs, ornaments, motifs in textiles in collaboration with home industries are produced
including some similarity objects in natures, e.g. 2 ornaments are shown in Fig. 4 according
to the obtained design as innovative activities in calculus and complex function.

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Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

Figure 3. High school students activity for creating puzzle and lampshade (Parhusip,2016) from a pair of

parametric equation (Eq.1) mapped by f(z)= exp(z).

Figure 4. Left(above): design from a parametric curve mapped by a complex function (called Rocked Math) ;
(left, below)
ornament as a souvenir from a silver and a garden decoration (right)
(Parhusip,2014)(Parhusip,2015).

2.3 The used of functions for data modeling
Typical problem arises from mathematics students is how to deal with data using
calculus. Most people refer to employ statistics for data analysis. Unfortunately, some data
need to be modeled into mathematical expression before to be analyzed. In classroom,
students mostly are introduced functions without any data in hands. By research approach,
students are encouraged to model data into an appropriate function leading to optimization
problems. Several data from local companies and laboratories in Indonesia have been
collected and discussed to give knowledge for beginners in this direction (Parhusip,2016).
This approach will not be explored into detail here.
Finally, the method for transferring creativities involve in curriculum must also be
taken into account since a standard curriculum requires rigidity in contents. In the section of
result and discussion, the activities and the results are explored to inspire readers.

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Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

3.

METHODOLOGY

3.1 Plane geometry into innovated platonic solids
We have known that regular plane geometry is normally introduced for students in
middle schools until undergraduate, e.g. triangle, rectangle and square, other regular
polygon and including circle. Only regular triangle, square and pentagon can be defined into
platonic solid yielding to only 5 kinds of platonic solid, i.e. tetrahedron, octahedron,
icosahedrons, cube and dodecahedron. The first 3 platonic solids characterized by 3, 4, 5
triangles meet at each vertex respectively where as a cube by 3 squares and dodecahedron by 3
pentagons meet at each vertex. By learning angles at each vertex must be less than 360 0 , one
may obtain only 5 platonic solids.

The innovation is devoted to make a special dress with dodecahedron patterns. Drawing
and repeating for several patterns of dodecahedron with papers and sticking textiles on these
patterns as depicted in Fig.5. Finally, a number of dodecahedrons are used.

Figure 5. Patterns of dodecahedron for special dress

3.2 Plane geometry for Leonardo Dome and Leonardo Bridge
Though only rectangles, we may have several activities for developing materials for
students to play with geometry. For instance, using bamboos as local materials in Indonesia
to create educative activities referring to Rinus Reolofs. Two typical examples here are
Leonardo Dome and Leonardo Bridge. The construction of both models are unique,i.e.
without using any glue, one may build the models. Additionally physical laws can be
explained through these educative materials.
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RESULTS AND DISCUSSION

4.1 Geometrics activity for middle school students and the related mathematical topics
As stated above, students must engage into an activity to involve the curriculum. In
this level, students should learn about plane geometry including traditional solid objects

such as a cylinder, sphere. Students have been introduced with several traditional steps :
the basic knowledge with related topic, i.e. a triangle, a rectangle, a circle, a cylinder, and a
sphere and computation for each geometrical object with new approach. In both activities,
students are involved in their own discoveries guided by mentors. Some photos of these
activities are shown in Fig.4. One of examples of the material is known as Tanggram.
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Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

Figure 4. Transferring basic knowledge of plane geometry by involving students to create their own materials
(Source :private photo, students are from middle schools of Santa Ursula BSD Jakarta and BPK Penabur Cirebon
Indonesia under the project Junior Summer Camp, in 10-19 June 2017, Salatiga, Indonesia).

The innovation is then dealt with some local materials leading to other special
keywords in mathematics and physics. Software called Surfer is used to introduce algebraic
surfaces and the innovation. Cylinder, ellipsoid and ball are typical algebraic surfaces known
by middle or high schools students. However combinations of those surfaces have not been

discussed and difficult to be illustrated manually. Using a software, some possible other
surfaces are easily introduced without learning complete related mathematical terms.
Therefore students are introduced to work with Surfer and applied into local culture activity,
i.e. create batik motifs according to the drawing results and partly the results are depicted in
Fig 4 (Parhusip,2017)

Figure 4. Students are creating and painting motifs from Surfer (left) and the related batik (right) (Source
:private photo, students are from middle school of Santa Ursula BSD Jakarta and BPK Penabur Cirebon
Indonesia under the project Junior Summer Camp, in 10-19 June 2017,Salatiga, Indonesia June 2017)

Geometrical innovation is then developed with the material adopted from Rinus Reolofs, i.e.
building Leonardo Dome and Leonardo Bridge from woods. Before this activity is presented,
an architecture need in future could be introduced such that students are motivated to
involve.

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Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

Figure 5. Building Leonardo Dome (left) and Leonardo Bridge (right) (Source :private photo, students are from
middle schools of Santa Ursula BSD Jakarta and BPK Penabur Cirebon Indonesia under the project Junior
Summer Camp, in 10-19 June 2017,Salatiga, Indonesia June 2017).

4.2 Activities in undergraduate students and the related mathematical topics
Fortunately, the same topics for middle schools can also be designed for undergraduate
students though the formal mathematical contents must be deeper in this case.
Undergraduate students are learning platonic solids and algebraic surfaces in formal
manners. Finally, students must find new performances to presents these topics.
4.2.1 Parametric curve into curves stitching and surfaces
One of small topics in physics and mathematics is learning parametric curves
including the properties. In this research, a modified hypocycloid curve is innovated into
many kinds of curves that one may use into ornaments. An old version of the activity is
doing curves stitching by representing curves with threads. The same curves can also be
ornamented using iron as shown in Fig.6. The variations of curves are obtained due to
different values of parameters in the parametric equations. Additionally every curve has its
derivative yielding to a new curve. Therefore this activity is also revealing an operator
mathematics such as derivative into an object.

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Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

Figure 6. A modified hypocycloid curve is defined into several curves stitching and ornaments .

The same parametric equation can also be extended into a surface with spherical
coordinates and innovated with other mathematical terms such as Golden ratio, Fibonacci
sequence to employ in the equations. Many surfaces can be obtained and some of these are
depicted in Fig. 7. Mathematical formulations of surfaces have been presented into detail
(Parhusip,2015).

Figure 7. Innovative surfaces from a modified hypocycloid curve

The same designs of surfaces are able to be other applications, e.g. motifs for textiles
called batik, designs for motifs in bags, souvenirs, and puzzles.
4.2.2 Geometrical innovations with Rinus Reolofs styles
Plane geometry is one of topics for schools and undergraduate mathematics
students. Innovations have been available in the internet with materials in traditional
mathematical questions added with modern presentations such as on line worksheets, using
software. These methods sometimes are not known by students and educators in villages in
Indonesia. Therefore mobile materials may be suitable choices for students to have
interaction between the given topic and an innovation material. The plane geometry is
innovated into platonic solids for instance where students may use papers to construct new
performance of platonic solids. Other possible material is dealing with outdoor activity by
creating Leonardo Dome and Leonardo Bridge as relations between geometrical topic with
architecture and physics. Referring to Rinus Reolofs, Leonardo Dome is created using ice
sticks and bamboos as the local materials in Indonesia and Leonardo Bridge may be built
using woods which are save materials for students and depicted in Fig. 8.
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Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

Figure 8. Models of Leonardo Dome using stick ices (left), wood (middle) and bamboo
(right)
4.2.3 Innovation with unused materials
Students may learn triangles and higher degrees of polygon. The pattern can be used for
several media using unused material such as bottle aqua. Figure 9 depicts the model of
hexagon.

Figure 9. Hexagon patterns are used to create for innovative ODEMA(Ornament Decorative
Mathematics)

4.2.4 Algebraic Surfaces into Local Cultures in textiles motifs in Batik
Students may also learn the same topic for middle schools such as learning cylinder
and sphere as regular algebraic surfaces introduced in classroom. Improving knowledge in
algebraic surfaces, students are able to work the surfaces into motifs of textiles called batik
9

Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

as local culture in Indonesia. Using Surfer as friendly software, students create a lot of
motifs. Guarded by a mentor from a home industry in Batik, students finally have their own
innovation in these algebraic surfaces. Some results are shown in Fig.9.

Figure 9. Painting textiles (Batik) with algebraic surfaces as motifs from Surfer done by undergraduate
mathematics students UKSW, February 2017.

CONCLUSION
Several approaches on innovating mathematics are presented here due to research
activities done for creating new media for promoting mathematics, geometry particularly.
The materials are defined due to the level of students: middle and junior schools and
undergraduate students. Platonic solids created from regular planes are innovated into
dress or other ornaments following Rinus Reolofs’s patterns. In the level of undergraduate
students, parametric curves and complex functions are explored into innovative curves and
surfaces being new to be new curves for curves stitching, motifs in several kinds of
souvenirs and accessories.
Algebraic surfaces are introduced using Surfer for all level students through batik
painting activity known to be a usual interesting activity for students offered in schools and
universities. This activity becomes a combination between local culture and transferring
new technology for students and home industries.

REFERENCES
Mahajan,S.,2014. Mathematics for 21st Century, Center for Redesign Curriculum,
Boston,Massachusetts.
Web 1. 2012. Fueling Innovation and Discovery, The Mathematical Sciences, in the 21st
Century , https://www.nap.edu/read/13373/chapter/1

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Presented on international conference on October 19 – 21, 2017 in the DLSU-D,Ca ite, Philippi es, ith the the e Emerging Trends
and Challenges in Mathematics Education .
Organized by the Philippine Council of Mathematics Teacher Educators, (MATHTED) Inc.
in partnership with the Mathematics and Statistics Department (MSD) of De La Salle University - Dasmariñas (DLSU-D)

Web2. https://materia.nl/article/mathematics-inspired-design-elevation-by-rinus-roelofs/
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1
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 z
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

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